HP 15c - Manual (Page 5)

Contents 5
Applications ......................................................................................................................... 65
Storing and Recalling Complex Numbers Using a Matrix .............................................. 65
Calculating the nth Roots of a Complex Number ............................................................ 67
Solving an Equation for Its Complex Roots .................................................................... 69
Contour Integrals .............................................................................................................. 73
Complex Potentials .......................................................................................................... 76
Section 4: Using Matrix Operations ................................................................ 82
Understanding the LU Decomposition ................................................................................ 82
ILL-Conditioned Matrices and the Condition Number ....................................................... 84
The Accuracy of Numerical Solutions to Linear Systems .................................................. 87
Making Difficult Equations Easier ...................................................................................... 88
Scaling .............................................................................................................................. 88
Preconditioning ................................................................................................................ 91
Least-Squares Calculations.................................................................................................. 93
Normal Equations ............................................................................................................. 93
Orthogonal Factorization.................................................................................................. 95
Singular and Nearly Singular Matrices ............................................................................... 98
Applications ....................................................................................................................... 100
Constructing an Identity Matrix ..................................................................................... 100
One-Step Residual Correction ........................................................................................ 101
Solving a System of Nonlinear Equations ..................................................................... 102
Solving a Large System of Complex Equations............................................................. 108
Least-Squares Using Normal Equations ........................................................................ 111
Least-Squares Using Successive Rows .......................................................................... 118
Eigenvalues of a Symmetric Real Matrix ...................................................................... 125
Eigenvectors of a Symmetric Real Matrix ..................................................................... 130
Optimization ................................................................................................................... 135
Appendix: Accuracy of Numerical Calculations ......................................... 145
Misconceptions About Errors ............................................................................................ 145
A Hierarchy of Errors ........................................................................................................ 150
Level 0: No Error ............................................................................................................... 150
Level ∞: Overflow/Underflow ......................................................................................... 150
Level 1: Correctly Rounded, or Nearly So ........................................................................ 150
Level 1C: Complex Level 1............................................................................................... 153
Level 2: Correctly Rounded for Possibly Perturbed Input ................................................ 154
Trigonometric Functions of Real Radian Angles ........................................................... 154
Backward Error Analysis ............................................................................................... 157
Backward Error Analysis Versus Singularities .............................................................. 161
Summary to Here ........................................................................................................... 162
Backward Error Analysis of Matrix Inversion ............................................................... 168
Is Backward Error Analysis a Good Idea? ..................................................................... 171
Index ................................................................................................................. 178